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Vector- and Tensor-Valued Descriptors for Spatial Patterns

  • Claus Beisbart
  • Robert Dahlke
  • Klaus Mecke
  • Herbert Wagner
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 600)

Abstract

Higher-rank Minkowski valuations are efficient means of describing the geometry and connectivity of spatial patterns. We show how to extend the framework of the scalar Minkowski functionals to vector- and tensor-valued measures. The versatility of these measures is demonstrated by using simple toy models as well as real data.

Keywords

Convex Body Spiral Galaxy Galaxy Cluster Antibonding State Minkowski Functional 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abraham, R. G., F. Valdes, H. K. C. Yee, S. van den Bergh (1994): ‘The Morphologies of Distant Galaxies. I. An Automated Classification System’. Ap. J. 432, pp. 75–90.CrossRefADSGoogle Scholar
  2. 2.
    Abraham, R. G., S. van den Bergh, K. Glazebrook, R. S. Ellis, B. X. Santiago, P. Surma, R. E. Griffiths (1996): ‘The Morphologies of Distant Galaxies. II. Classifications from the Hubble Space Telescope Medium Deep Survey’. Ap. J. Suppl. 107, pp. 1.CrossRefADSGoogle Scholar
  3. 3.
    Alesker, S. (1999): ‘Continuous Rotation Invariant Valuations on Convex Sets’. Ann. of Math. 149(3), p. 977.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alesker, S. (1999): ‘Description of Continuous Isometry Covariant Valuations on Convex Sets’. Geom. Dedicata 74, p. 241.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bader, R. F.W. (1990): Atoms in Molecules — A Quantum Theory. (Oxford University Press, Oxford).Google Scholar
  6. 6.
    Bartelmann, M., A. Huss, J. M. Colberg, A. Jenkins, F. R. Pearce (1998): ‘Arc Statistics with Realistic Cluster Potentials. IV. Clusters in Different Cosmologies’. Astron. Astrophys. 330, pp. 1–9.ADSGoogle Scholar
  7. 7.
    Beisbart, C. (2001): Measuring Cosmic Structure. Minkowski Valuations and Mark Correlations for Cosmological Morphometry. Ph.D. thesis, Ludwig-Maximilians-Universität München.Google Scholar
  8. 8.
    Beisbart, C., T. Buchert, H. Wagner (2001): ‘Morphometry of Spatial Patterns’. Physica A 293/3–4, p. 592.zbMATHCrossRefADSGoogle Scholar
  9. 9.
    Beisbart, C., K. Mecke (2002): ‘Tensor Valuations’. In preparation.Google Scholar
  10. 10.
    Beisbart, C., R. Valdarnini, T. Buchert (2001): ‘The Morphological and Dynamical Evolution of Simulated Galaxy Clusters’. Astron. Astrophys. 379, pp. 412–425.CrossRefADSGoogle Scholar
  11. 11.
    Block, D. L., I. Puerari, R. J. Buta, R. Abraham, M. Takamiya, A. Stockton (2001): ‘The Duality of Spiral Structure, and a Quantitative Dust Penetrated Morphological Tuning Fork at Low and High Redshift’. In’ ASP Conf. Ser. 230: Galaxy Disks and Disk Galaxies’.Google Scholar
  12. 12.
    Cohen-Tannoudji, C., B. Diu, F. Laloë (1977): Quantum Mechanics: Volume II. (John Wiley & Sons, Chichester), 2nd edition.Google Scholar
  13. 13.
    Dahlke, R. (2000), ‘Dynamische Perkolationsmodelle zur Morphologie von Galaxien’, Diploma thesis, Ludwig-Maximilians-Universität München.Google Scholar
  14. 14.
    Fischer, P., G. Bernstein, G. Rhee, J. A. Tyson (1997): ‘The Mass Distribution of the Cluster 0957+561 from Gravitational Lensing’. A. J. 113, p. 521.CrossRefADSGoogle Scholar
  15. 15.
    Gerola, H., P. E. Seiden (1978): ‘Stochastic Star Formation and Spiral Structure of Galaxies’. Ap. J. 223, pp. 129–135.CrossRefADSGoogle Scholar
  16. 16.
    Hadwiger, H. (1957): Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. (Springer Verlag, Berlin).zbMATHGoogle Scholar
  17. 17.
    Hadwiger, H., C. Meier (1974): ‘Studien zur vektoriellen Integralgeometrie’. Math. Nachr. 56, p. 361.MathSciNetGoogle Scholar
  18. 18.
    Hadwiger, H., R. Schneider (1971): ‘Vektorielle Integralgeometrie’. Elemente der Mathematik 26, p. 49.zbMATHMathSciNetGoogle Scholar
  19. 19.
    Hubble, E. (1926): ‘Extra-galactic Nebulae’. Ap. J. 64, p. 321.CrossRefADSGoogle Scholar
  20. 20.
    Kerscher, M. (2000): ‘Statistical analysis of large-scale structure in the Universe’. In: Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, ed. by K. R. Mecke, D. Stoyan, Number 554 in Lecture Notes in Physics (Springer, Berlin), astro-ph/9912329.Google Scholar
  21. 21.
    Klain, D. A. (1995): ‘A Short Proof of Hadwiger’s Characterization Theorem’. Mathematika 42, pp. 329–339.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Kornreich, D. A., M. P. Haynes, R. V. E. Lovelace, L. van Zee (July 2000): ‘Departures From Axisymmetric Morphology and Dynamics in Spiral Galaxies’. A. J. 120, pp. 139–164.CrossRefADSGoogle Scholar
  23. 23.
    Lin, C. C., F. H. Shu (1964): ‘On the Spiral structure of Disk Galaxies’. Ap. J. 140, p. 646.CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    McMullen, P. (1997): ‘Isometry Covariant Valuations on Convex Bodies’. Rend. Circ. Mat. Palermo (2) 50, pp. 259–271.MathSciNetGoogle Scholar
  25. 25.
    Mecke, K. (2000): ‘Additivity, Convexity, and beyond: Application of Minkowski Functionals in Statistical Physics’. In: Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, ed. by K. R. Mecke, D. Stoyan, Number 554 in Lecture Notes in Physics (Springer, Berlin).Google Scholar
  26. 26.
    Mecke, K. R. (2002): ‘Fundamental Measure Density Functional for Mixtures of Nonspherical Hard Particles’. In preparation.Google Scholar
  27. 27.
    Mohr, J. J., D. G. Fabricant, M. J. Geller (1993): ‘An x-ray-method for detecting substructure in galaxy clusters’. Ap. J. 413, p. 492.CrossRefADSGoogle Scholar
  28. 28.
    Mohr, J. J., A. E. Evrard, D. E. Fabricant, M. J. Geller (1995): ‘Cosmological Constraints from X-ray Cluster Morphologies’. Ap. J. 447, p. 8.CrossRefADSGoogle Scholar
  29. 29.
    Naim, A., O. Lahav, R. J. Buta, H. G. Corwin, G. de Vaucouleurs, A. Dressler, J. P. Huchra, S. van den Bergh, S. Raychaudhury, L. Sodre, M. C. Storrie-Lombardi (1995): ‘A Comparative Study of Morphological Classifications of APM Galaxies’. Mon. Not. R. Astron. Soc. 274, pp. 1107–1125.ADSGoogle Scholar
  30. 30.
    Peacock, J. (1999): Cosmological Physics. (Cambridge University Press, Cambridge).zbMATHGoogle Scholar
  31. 31.
    Peebles, P. J. E. (1993): Principles of Physical Cosmology. (Princeton University Press, Princeton, New Jersey).Google Scholar
  32. 32.
    Reiss, H., H. Frisch, J. L. Lebowitz (1959): ‘Statistical Mechanics of Rigid Spheres’. J. Chem. Phys. 31, pp. 369–380.CrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Richstone, D., A. Loeb, E. L. Turner (July 1992): ‘A Lower Limit of the Cosmic Mean Density from the Ages of Clusters of Galaxies’. Ap. J. 393, p. 477.CrossRefADSGoogle Scholar
  34. 34.
    Rosenfeld, Y. (1989): ‘Free-energy Model for the Inhomogeneous Hard-sphere Fluid Mixture and Density-functional Theory of Freezing’, Phys. Rev. Lett. 63, pp. 980–983.CrossRefADSGoogle Scholar
  35. 35.
    Rosenfeld, Y. (1994): ‘Density Functional Theory of Molecular Fluids: Free-energy Model for the Inhomogeneous Hard-body Fluid’. Phys. Rev. E 50, pp. R3318–3321.CrossRefADSGoogle Scholar
  36. 36.
    Roth, R. (1999): Depletion Forces in Hard Sphere Mixtures. Ph.D. thesis, University Wuppertal, (WUB-DIS 99-19).Google Scholar
  37. 37.
    Russell, W. S., W.W. Roberts (September 1993): ‘Analysis of the Distribution of Pitch Angles in Model Galactic Disks — Numerical Methods and Algorithms’. Ap. J. 414, pp. 86–97.CrossRefADSGoogle Scholar
  38. 38.
    Schmalzing, J., T. Buchert (1997): ‘Beyond Genus Statistics: a Unifying Approach to the Morphology of Cosmic Structure’. Ap. J. Lett. 482, p. L1.CrossRefADSGoogle Scholar
  39. 39.
    Schmalzing, J., T. Buchert, A. L. Melott, V. Sahni, B. S. Sathyaprakash, S. F. Shandarin (1999): ‘Disentangling the Cosmic Web. I. Morphology of Isodensity Contours’. Ap. J. 526, p. 568.CrossRefADSGoogle Scholar
  40. 40.
    Schneider, P., J. Ehlers, E. E. Falco (1992): Gravitational Lenses. (Springer-Verlag, Berlin. Also Astronomy and Astrophysics Library).CrossRefGoogle Scholar
  41. 41.
    Schneider, R. (1972): ‘Krümmungsschwerpunkte konvexer Körper (I)’. Abh. Math. Sem. Univ. Hamburg 37, p. 112.zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Schneider, R. (1972): ‘Krümmungsschwerpunkte konvexer Körper (II)’. Abh. Math. Sem. Univ. Hamburg 37, p. 202.Google Scholar
  43. 43.
    Schneider, R. (1993): Convex Bodies: the Brunn-Minkowski Theory. (Cambridge University Press, Cambridge).zbMATHCrossRefGoogle Scholar
  44. 44.
    Schneider, R. (2000): ‘Tensor Valuations on Convex Bodies and Integral Geometry’. Rend. Circ. Mat. Palermo, Ser. II, Suppl. 65, p. 295.Google Scholar
  45. 45.
    Schneider, R., R. Schuster (2001): ‘Tensor Valuations on Convex Bodies and Integral Geometry, II’. In preparation.Google Scholar
  46. 46.
    Seiden, P. E., H. Gerola (1982): ‘Propagating Star Formation and the Structure and Evolution of Galaxies’. Fundamentals of Cosmic Physics 7, pp. 241–4311.ADSGoogle Scholar
  47. 47.
    Serra, J. (ed.) (1994): Mathematical Morphology and its Applications to Image Processing (Kluwer, Dordrecht).zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Claus Beisbart
    • 1
    • 2
  • Robert Dahlke
    • 2
  • Klaus Mecke
    • 3
    • 4
  • Herbert Wagner
    • 2
  1. 1.Nuclear & Astrophysics LaboratoryUniversity of OxfordOxfordUK
  2. 2.Sektion PhysikLudwig-Maximilians-UniversitätMünchenGermany
  3. 3.Max-Planck-Institut für MetallforschungStuttgartGermany
  4. 4.Institut für Theoretische und Angewandte Physik, Fakultät für PhysikUniversität StuttgartStuttgartGermany

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